[Paper Review] On Multi Poly-Bernoulli Polynomials

Roberto B. Corcino, Hassan Jolany+2 more|arXiv (Cornell University)|Jul 13, 2016

Advanced Mathematical Identities12 references3 citations

TL;DR

This paper introduces generalized multi poly-Bernoulli polynomials using multiple polylogarithms and derives an explicit formula for Hurwitz-Lerch type multi poly-Bernoulli polynomials in terms of r-Whitney numbers of the second kind. The key contribution is a closed-form expression linking these polynomials to combinatorial number theory via special functions and Stirling-type numbers.

ABSTRACT

In this paper, we define multi poly-Bernoulli polynomials using multiple polylogarithm and derive some properties parallel to those of poly-Bernoulli polynomials. Furthermore, an explicit formula for certain Hurwitz-Lerch type multi poly-Bernoulli polynomials is established using the $r$-Whitney numbers of the second kind.

Motivation & Objective

• To extend poly-Bernoulli polynomials to a multi-parameter setting using multiple polylogarithms.
• To define Hurwitz-Lerch type multi poly-Bernoulli polynomials using generalized zeta functions.
• To establish an explicit formula for these polynomials in terms of r-Whitney numbers of the second kind.
• To provide combinatorial interpretations for special cases when parameters are negative integers.

Proposed method

• Define generalized multi poly-Bernoulli polynomials via the generating function involving multiple polylogarithms and exponential terms.
• Introduce the Hurwitz-Lerch type multi poly-Bernoulli polynomials using the generating function involving the generalized Hurwitz-Lerch multiple zeta function.
• Utilize the exponential generating function of r-Whitney numbers of the second kind to derive the explicit formula.
• Express the generating function of the polynomials as a series involving (1−e⁻ᵗ) and exponential terms to connect to the r-Whitney numbers.
• Establish a connection between the r-Whitney numbers and Stirling numbers of the second kind through the identity W₋₁,ₓᵣ(n, mᵣ) = (−1)ⁿ⁺ᵐʳ S(n, mᵣ).
• Use the generating function of the Hurwitz-Lerch multiple zeta function to derive the explicit formula for the polynomials.

Experimental results

Research questions

• RQ1 How can poly-Bernoulli polynomials be generalized to multiple parameters using multiple polylogarithms?
• RQ2 What is the explicit formula for Hurwitz-Lerch type multi poly-Bernoulli polynomials in terms of known combinatorial numbers?
• RQ3 How do r-Whitney numbers of the second kind relate to the coefficients in the generating function of these polynomials?
• RQ4 Can non-negative integer values of the polynomials for negative parameters be given a combinatorial interpretation?

Key findings

• The explicit formula for Hurwitz-Lerch type multi poly-Bernoulli polynomials is given by: Bₙ,ₐ⁽ᵏ¹,…,ᵏʳ⁾(x) = Σ₀≤ₘ₁≤…≤ₘʳ≤ₙ [mᵣ! W₋₁,ₓᵣ(n, mᵣ)] / [(m₁+a−r+1)ᵏ¹ ⋯ (mᵣ+a)ᵏʳ].
• When x = 0, the formula reduces to Bₙ,ₐ⁽ᵏ¹,…,ᵏʳ⁾ = Σ₀≤ₘ₁≤…≤ₘʳ≤ₙ [(-1)ⁿ⁺ᵐʳ mᵣ! S(n, mᵣ)] / [(m₁+a−r+1)ᵏ¹ ⋯ (mᵣ+a)ᵏʳ], where S(n, mᵣ) denotes Stirling numbers of the second kind.
• The result recovers the known explicit formula for Hurwitz-Lerch poly-Bernoulli numbers when r = 1.
• For negative integer parameters k₁, ..., kᵣ, the values Bₙ,ₐ⁽⁻ᵏ¹,…,⁻ᵏʳ⁾ are non-negative integers, suggesting potential combinatorial interpretations.
• The generating function of the r-Whitney numbers of the second kind is used to derive the main formula, linking special functions to combinatorial sequences.
• The connection between r-Whitney numbers and Stirling numbers is confirmed via the identity W₋₁,₀(n, mᵣ) = (−1)ⁿ⁺ᵐʳ S(n, mᵣ).